Contents

Preface Lizhen Ji, Frans Oort, and Shing-Tung Yau ...... ix Photographs ...... xi

The Legacy of Bernhard After One Hundred and Fifty Years, Volume I What One Should Know About Riemann But May Not Know? Lizhen Ji and Shing-Tung Yau ...... 1 Riemann’s Influence in Geometry, Analysis and Number Theory Sir Michael F. Atiyah...... 57 Riemann’s Saddle-point Method and the Riemann–Siegel Formula M. V. Berry ...... 69 The Period Matrices and Theta Functions of Riemann Ching-Li Chai ...... 79 Riemann’s Hypothesis Brian Conrey ...... 107 Extension of Holomorphic Functions Defined on Non-Reduced Analytic Subvarieties Jean-Pierre Demailly...... 191 Bundles with Extra Geometric or Dynamic Structure F. T. Farrell...... 223 The Theory of Shock Waves: From Riemann through Today James Glimm, Dan Marchesin, and Bradley Plohr...... 251 Riemann’s Existence Theorem David Harbater ...... 275 The Historical Roots of the Concept of Riemann Surfaces Lizhen Ji...... 287 The Story of Riemann’s Moduli Space Lizhen Ji...... 307 Riemann and the Modern Concept of Space Jürgen Jost ...... 359

vii viii CONTENTS

The Legacy of After One Hundred and Fifty Years, Volume II Survey on Analytic and Topological Torsion Wolfgang Lück...... 379 The Riemann Minimal Examples William H. Meeks III and Joaquín Pérez...... 417 Manifolds of Mappings and Shapes Peter W. Michor...... 459 The over Finite Fields: From Weil to the Present Day James S. Milne ...... 487 The Riemann-Hurwitz Formula Frans Oort ...... 567 Early History of the Riemann Hypothesis in Positive Characteristic Frans Oort and Norbert Schappacher ...... 595 Riemann’s Influence in Number Theory from a Computational and Experimental Perspective Michael Rubinstein...... 633 Riemann Problem and Shock Capturing Schemes Chi-Wang Shu ...... 681 A Discourse on the Measurable & Incompressible Fluid Motion Dennis Sullivan...... 691 The Mertens Conjecture Herman J. J. te Riele ...... 703 Hodge Structures, Coniveau and Algebraic Cycles Claire Voisin...... 719